Question 16
Let $N(a, b)$ be the number of ways to cover an $a \times b$ board using domino tiles. Additionally, let $N^*(a, 2b + 1)$ be the number of ways to cover an $a \times (2b+1)$ board using domino tiles, without having vertical dominoes in the central column. Prove that $N^*(2m, 2n + 1) = 2^m N(2m, n)N(2m, n - 1)$.